the use of imaginary characters in the summation of series, proceed-
ing according to the powers of the series and cosines of arcs in
arithmetical progression. He likewise shows, that according to his
mode of explanation, certain ambiguous expressions that occur in
analysis are perfectly intelligible, and that in the controversy con-
cerning the logarithms of negative quantities, carried on formerly
between Leibnitz, Bernoulli, Euler, and Dalembert, all paradox and
ambiguity may be made to disappear, by referring to the origin and
real import of the impossible exponential quantities.
Although the principal object of this paper is to vindicate the in- dubitable justness of the operation conducted with imaginary cha- racters, yet in the latter part some arguments are likewise offered in favour of the commodiousness of imaginary expressions for facilitating calculations. And, lastly, it is contended, that in the present state of analysis, these expressions are particularly useful in deducing certain conclusions, which without their aid could not be obtained without much difficulty
By Mr. Richard Walker. Communicated by Henry Cavendish, Esq
F.R.S. Read January 22, 1801. [Phil. Trans. 1801, p. 120.Mr. Walker, since his late communications to the Society on the best means of producing artificial cold, received intelligence that Mr. Lowitz, Professor of Chemistry at Petersburgh, had made some experiments, in which a neutral salt different from those he had him- self used, and which is but little known or attended to, produced effects which exceeded his expectations. The salt is the muriated lime; which, mixed with snow in the proportion of about 3 to 2, at the temperature of +27°, produced a refrigeration which sunk the ther- mometer to-55; and with this mixture the Professor in one ex- periment froze no less than 351bs of quicksilver.
Mr. Walker repeated the experiment with success; but finding that it can only be made during a freezing atmosphere, he resolved to try the effect of this salt, reduced to such a strength by evapora- tion as to endure being kept in a solid state throughout the year. After describing the expedients used for this purpose, he enumerates the results of two sets of experiments; the first made with the mu riated lime prepared so as to be used in winter only, that is, of the specific strength of 1 450; and the other made with the salt prepared so as to be kept for use at any time, the strength of which was 1-490 The apparatus here used (tbough somewhat improved) is not unlike that described in Mr. Walker's former communication, nor is the process materially different.
The раper concludes with a general view of the different frigorific mixtures:—1st, those composed of chemical substances with ice; and 2nd, those in which the use of ice is dispensed with. In a postscript the author suggests a method of obtaining transparent ice, fit for optical purposes, which is effected merely by immersing a vessel
