VOL. LXXXI.] PHILOSOPHICAL TRANSACTIONS. 87 



Let IT be the general term, then resolve '^ into an infinite series, and take the 

 fluent on both sides; then write 2, 3, 4, &c. for x, and one side will become 

 the hyp. log. of the given series, and the value of the other side may be found 

 from the tables. 



Ex. J. To find the value of ^ X | X ^r X &c. ad infinitum. 



Here the general term is ^_^; hence - = — ——^. = — );i — ^ — 1^ — 

 &c.; hence the hyp. "log. — ^ = Jz + jpf + ^ + &c. Write 2, 3, 4, &c. 



for X, and we have the hyp. log. - + hyp. log. - + hyp. log. — + &c. = a + 

 ic + iE + &c. = .693 147 180574, which is the hyp. log. 2; but hyp. log. 

 J + hyp. log. I + hyp. log. i^ + &c. = hyp. log. ^ X | X || X &c. con- 

 sequently i X ^ X i| X &c. = 2. 



Ex. 2. To find the value of - X 4 X tt- X &c. ad infinitum. 



7 20 o3 



-J , 1 .. • f . '^ 3i- 3.V 3ji- 3* 



Here the general term is ; hence - =: := to — 



&c.; hence the hyp. log. — — = - -f -i- -i- ^ + &c. Write 2, 3, 4, &c. 

 for 0,', and we have the hyp. log. - + hyp. log. ^, + hyp. log. ^ + &c. = b -f 



o 07 (>4. 



iE + ^H -}- &c. = .211466250444; or hyp. log. ^ X -• X ^ X &c. = 



.211466250444; hence ^- X ^ X ^ X &c. = 1.6272^5, &c. 



Hence we may find the value of such a quantity, supposing the number of 

 factors to be finite. 



XXI. Experiments and Observations to Investigate the Composition of Jameses 

 Powder. By George Pearson, M. D., F. R. S. p. 317- 

 Dr. P. after remarking that James's Powder, on which many physicians prin- 

 cipally depend in the cure of continued fevers, was originally a patent medicine ; 

 but that it is well-known that it cannot be prepared by following the directions 

 of the specification in the court of chancery ; proceeds to state the sensible pro- 

 perties of this powder, its specific gravity, the effects of high degrees of heat 

 upon it, and the action of diflerent menstrua to which it was subjected. From 

 these analytic experiments he infers that the proportions of the several consti- 

 tuent parts of this powder are as follow : 



Grains. 



Phosphorated lime, with a little antimonial calx 100. 



Algaroth powder 37. 1 5 



Insoluble antimonial calx, with a little phosphorated lime IP-S 5 



The same insoluble calx, with, probably, a little phosphorated lime 55. 



Waste 8. 



240.0 



