no 



Prof. PowelPs Abstract ofM. Cauchy's 



breviation, writing the sums of the coefficients derived from 

 those terras of (li^.) which involve cos^ a, cos* /3, cos^ y, re- 

 spectively L, M, N ; and those which involve cos j3 cos y, 

 cos y cos a, cos a cos /3, respectively P, Q, R, those equations 

 are reduced to the following : 



dt^ 



dt^ 

 d^ 

 dt^ 



= -[R0 + M,,+P?]> 



= - [Q0+P>j + N?] 



(22.) 



These equations enable us at the end of the time /, to deter- 

 mine the three functions f >) ? ; which is, in fact, done, if we 

 have determined the six functions d e f g h i; and this we can 

 effect by means of the initial values of these functions, and their 

 differential coefficients with respect to t. Writing these initial 

 values of the functions by subjoining (o), and those of their dif- 

 ferential coefficients by subjoining (1.), we shall have by the 

 formula (17.), supposed reduced to a single term, 



00 = do cos kq + go sin ^ ^ ^ 

 >3o = Cq cos ^ ^ + ko sin k § 

 ^ = fo cos kg -\- io sin kg 



01 = dj cos kg + gi sin ^ ^ 

 >ji = Cj cos kg -f ki sin^^ 



> 



(23.) 



?i = fj COS kg + i, sin ^ ^ 



In order, by means of these values corresponding to / = 0, 

 to deduce those corresponding to any value of ^, we must pro- 

 ceed to the following considerations relative to the coefficients. 



Let the arbitrary quantities A B C be assumed as the co- 

 sines of the angles which a certain line OA through the 

 origin forms with the positive semiaxes ; or in other words, so 

 that we have 



A^ + B^ + C^ = 1 



and the line O A is represented by the equations 



•^ _ ^ _ ^ 

 X "■ B^ ■" IT' 



Also, if we suppose 



8 = A0 + B)3+C?, 



the value of a will give the displacement of the molecule m 

 in a direction parallel to the line OA, and positive or negative 

 according to the direction of that line. 



/ 



(24.) 



(25.) 



(26.] 



