824 F U N F U by means of the sign of integration B-c/ ^ ; but the esta blishment of the properties of the function logarithm (assumed to be is to be determined by means of a given relation between x, qx, <t>ax <b&x . . ; iu particular the given relation may be between x, $x, <pax; this can be at once reduced to equations of finite differences ; for writing x - u n , <& = u n+ - l > we have u n ^ = au n , giving u n , and therefore also x, each of them as a function of n ; and then writing Qx=v n , Qax. will be the same function of n + l,
- =v v and the given relation is again an equation of finite
differences in n+1 , t> n , and n ; we have thus v n , = <t>x, as a function of n, that is, of x. As regards the equation M M+I = a n , considered in itself apart from what precedes, observe that this is satisfied by writing u n = a n (x), or the question of solving this equation of finite differences is in fact identical with that of finding the TO- th function o" (a;), where 0(0;) is a given function ofx. It of course depends on the form of a(x) whether this question admits of solution in any proper sense ; thus, for a function such as log x, the 7i-th logarithm is expressible in its original function log".*;, (=log log ... x), and not in any other form. But there are forms, for instance ^-~ , where the Ti-th function a"x is a c + dx function of the like form a n x = ^ ^ , in which the actual value can be expressed as a function of n ; if o be such a form, then <pa<$>~ 1 , whatever <p may be, is a like form, for we obviously have ((pa<f>~ l ) n = ^o"^" 1 . The determination of the n-th function is in fact a leading question in the calculus of functions. It is to be observed that considering the case of two variables, if for instance a(x, y) denote a given function of x, y, the notation a*(x, y) is altogether meaningless ; in order to generalize the question we require an extended notation wherein a single func tional symbol is used to denote two functions of the two variables, thus <}>(x, y) = a(x, y), &(x, y), a and given functions; writing from shortness x 1 = a(x, y), y l = P(x,y), then <p z (x,y) will denote ^(i-j, i/j), that is, two functions a^, i/j), &(x lt yj, say these are x sj 2/2 ! 3 ( x > y} will denote <p(x z , y 2 ), and so on, so that <p n (x, y) will have a determinate meaning. And the like is obviously the case in regard to any number of variables, the single functional symbol denoting in each case a set of functions equal in number to the variables. (A. CA.) FUNDS, FUNDING SYSTEM. See NATIONAL DEBT. FUNEN. See DENMARK. FUNERAL RITES, ceremonies attending the burial, burning, or disposal otherwise of the dead. The prevalent modes of disposal are very various. The rudest is that of certain nomadic tribes, who, from the exi gency of their wandering life, simply expose their dead, un less the custom of some tribesin modern Guinea be still ruder, who, like the ancient Ichthyophagi, throw their dead into the sea, and think they have thus got rid of corpse and ghost to gether. In exposing the dead, various rules prevail : some bave them where they die and move their own camp on to other hunting grounds; others, like the Wanyamwesi, carry them a little distance into the forest and leave them to be devoured by beasts of prey ; while others still, like the Kamtchadales, refine upon this practice, and keep special dogs for the purpose of consuming the dead, reconciling themselves to this custom by the doctrine that they who are eaten by dogs here will drive fine dogs in the other world. Probably the straits of. war explain the practice of the Latookas of Zanzibar, who bury their dead, but make it a rule to leave those slain in battle to be devoured by wild beasts where they lie. The custom of exposure has survived among cultured nations in the case of the Parsees, who bring their dead to certain round towers, called towers of silence" to be eaten by vultures v/hich make these towers their dwelling-places. There is evidence that a more startling mode of disposal still is sometimes practised. Certain tribes of the ancient Scythians are said to have eaten their dead, and Burton says that at Dahomey the body of a person killed by lightning is not buried like others, but is hacked in pieces and eaten by the priests. Skertchley says they do not actually eat it, but only make believe to do so. but the make-believe is probably a survival of what was onco the reality. Next to exposure, burial is the simplest method of dis posal, and it is the most widespread, being common alike to the lowest and the highest forms of culture. Burial likewise has its varying customs, from laying the body in natural holes or caves to erecting over it majestic tem ples. A common practice is simply to lay the body out on the ground and pile stones, or, as among the Moors, prickly thorns, over it, to keep off beasts. The use of the coffin is no sure test of culture, for while some of the rudest peoples lay the dead in a hollowed trunk or a boat, the Mahometan nations use nothing of the kind. Nor is the possession of fixed burying places a better index to the stage of develop ment, for while many primitive tribes have stated burial places, the more advanced Mexicans had none. Some American tribes bury their children in a separate graveyard from adults, and others bury them by the wayside that their souls may enter into persons passing by. Refinements upon ordinary burial are the practices of first burning the dead, or embalming them, or drying them on trees or artificial scaffolds, before burying them. That plain burial is the earlier custom, and that these other more elaborate and costly fashions are later eu- graftments upon it, seems to be proved by the fact that many nations adopt more than one custom, and that, while they only bury their women, they first burn or drj their men. The ancient Colchians suspended the corpses of men in trees, but their women they buried. The Gonds and Bhils of India, who have adopted under Hindu influ ences the practice of burning, still bury their women, while the Todas burn all now, except children the victims of in fanticide, whom they bury. The Muddikers, who bury, burn lepers, probably from sanitary reasons ; and certain tribes who burn bury the lightning-struck on the spot where they fall. The Kalmucks follow all modes : their usual custom is exposure ; but they also burn, or bury, or throw the corpse into the water, or under a heap of stones, or build a hut over it, according to what the priest declares to be most suitable to the condition of the deceased. Another point of interest in burying is the position of the body in the grave. Some bury their dead lying, others sitting, and there is a remarkable consensus of custom for the practice of laying the body east and west, sometimes with the head to the east and sometimes to the west. This custom is evidently due originally to solar symbolism, and the head is turned to the east or to the west, according as the dead are thought of in connexion with the sunrise, the reputed home of deity, or with the sunset, the reputed region of the dead. This practice, however, though nearly universal, is not absolutely so, for some tribes lay their dead north and south ; and others, like the Bongos, bury men with the face to the north and women with the face to the south ; while if one of the Wanyamwcsi in Africa happens to die abroad, he is buried facing his native village. The necessary act of disposing of the corpse has always been accompanied by ceremonies expressive (1) of affection for the deceased or grief for his loss ; (2) of present interest in and solicitude for his welfare ; (3) of a certain mysteri ous fear of him in his present state ; and (4) of affectionate remembrance of him. These ceremonies vary much under different changes of culture, yet have all the same central elements. Among primitive nations the most common ceremonial expressions of grief are simple exaggerations of the natural expressions of the emotions, a carelessness as to usual com forts, and a positive distracting agony. Fasting, neglecting the hair, wearing rags or sackcloth, sitting in ashes, daubing
