] 821 .] On the Expansion of the Functions f {x),J'{x, y)j ^c, 359 



to exert a decided influence on the ova of animals ; he placed 

 the eggs of different species of phaUcncc. in transparent and 

 opaque jars, when he uniformly found that those in the black jars 

 were first hatched ; he, therefore, concludes that light is preju- 

 dicial to the developement of the eg^: thus we find, says he, 

 that the eggs of many birds are furnished with an opaque shell, 

 as those of birds, and that if it be delicate, the parent deposits 

 them in dark and concealed places." Before any conclusion caa 

 be legitimately deduced from the experiments of Michelotti, it 

 ought to be shown that the temperature in both jars was the 

 same ; for unless this were established by actual experiment, 

 we might be inclined to draw an opposite inference from the 

 different radiating powers of black and white surfaces. 



(7*0 he continued.) 



Article V. 



On the Expansion of the Functions f{x), f {x, y),f{Xjyj x), 4'C- 

 By Mr. James Adams. 



(To the Editor of the Annals of Philosophi/ .) 



SIR., Stonehouse^ near Phj mouth, March 3, 1821. 



The usual method of expanding the functions /(a), y (x, y), 

 f{i,y, z),f{x,y, z, v), 5cc. being from their nature very trouble- 

 some, I beg to recommend the following, which, I trust, will be 

 found more convenient and expeditious ; the insertion of which 

 in the Annals of Philosophi/ will oblige. Sir, 



Your humble servant, 



James Adams. 



Problem 1. — It is required to develope /" (.i), or any function 

 of X. 



Letf(x) = u; then place the symbols d, d-, d^, Sec. succes- 

 sively before the character ti ; likewise the symbols d and A 

 successively before the assumed series x + x" + x' -f J^^ + &c. 

 together with the corresponding numerical coefficients in the 

 following manner : 



r-i~- ^ ^ + tAtt-^ ^ '■'^'^ + T-ir-?^— . Ax^ + &c. w^e shall thea 



I . dx I .2 dx^ 1 . 2 . 3 d x^ 



have/Cr + A j) = ?^+ j~^ A j + — — - A x^- + j 



dx ' \ .2dx' ' 1 . 2 . S d »* 



A x' -f 8cc. 



