112 M. Caiichy's View of the Ufidulatory Theory of hight. 

 Now, writing bq »! ^^r the initial values of a and -jv j we 



Oi Z 



have 



«o = A^o + B>3o + C?o (34..) 



»i = A^i + B>ji + C?i; (35.) 



or, substituting the values of ^q ^j, &c., from (23. )» these forms 



become 



8o= [cloA + eoB + foC]cos^§+[goA + hoB + ioC]sin^g 



= ^ (g) (36.) 



«! = [d,A + eiB + fiC]cos^^ + [giA + hiB + iiC]sinA'^ 



= ^ (§), (37.) 



using the last symbols to designate the forms of these func- 

 tions of q. 



The form of the expression (28.) shows us at once that its 

 integral must be a trigonometrical function, which it will easily 

 be seen must take the following form, involving as coefficients 

 the initial values 



sin 5^ 



8 = 80 cos St -^ 8i— , (38.) 



or, what is the same thing, 



3 = 8o cos st + 8i /o cos s t dt, (39.) 



If we here substitute the values of «q gj and the trigonome- 

 trical values of the resulting products, and also write 



we shall at length deduce an expression, which, in compari- 

 son with equations (36.) (37.), may, by the same notation, 

 be expressed thus, (carefully observing that no greater gene- 

 rality is implied than belongs to (38.) and (39.) ): 

 'OS {q -r ft t) ^- ts {g-^nt) 

 2 ~ 



^ n^ n{g-^nt)-^n{g-nt) ^^^ 



Jo 2 



(40.) 



the form being the same for each of the values «' a'' «'" corre- 

 sponding to the three positive values of 5-, involving respec- 

 tively n' ir a'" and A' M' A'", &c. If these values of b\ 

 &c , be substituted in equation (33.), we have ^ >5 ^ in func- 

 tions of § and /, which satisfy the two conditions of the values 

 ^0, &c., when ^ = 0, and of the equations (22.) for any value 

 of^. 



Also the velocity co of the molecule at the end of any time t 



