w= 



its Analogues, and the Calculus of Imaginaries. 279 



and its duplicate 



^A e V(»i 2 +% 2 +%> {cos^a? + m^y + m 3 z) 



+ i n . sin (m x x + m$ + m^z) } 



1 ?# e */{m?+m?+rr h *)w $ COS^X + m$ + m S z) 



— i K . sin (m^ -f m 2 y + m 3 z) } __ 



Again, as before, regarding A and B as arbitrary functions of 

 m 19 m 2 , m 3) we can throw these solutions into the following 

 shapes : — 



? • 



W) 



w=4 



W 



W: 



(«s)" 



'fff<&{m it m 2 , m 3 ) e m ^+ m *y +m * z 



cos 4/ (m* -f wi 2 2 + m 3 2 ) w . dm l dmc L dm 3 



+ 



sin \/ (mj 2 + m 2 2 + m 3 2 ) w . dm l dmc L dm 3 j 

 with its duplicate 



e^ w i 2+Wi 2 2+WI 3 2 ^ . dm x dm (l dm 3 

 + ' > 



/ffy?{tyi 9 m<i, m 3 ) sin {m-^x + m 2 y + m 3 z) 



eVimf+mf+m&w % d?n 1 dm 2 dm 3 ^ 



the limits of the integrals in both cases being supposed inde- 

 pendent of the quantities x, y, z and w. The same remark 

 which has been made upon the similar solutions of (I.) and (II.) 

 will apply here with equal force. 



By a modification of Poisson's solution of the equation of 

 oscillatory motion in an unlimited gas, we should obtain a solu- 

 tion of equation (III.) strictly in the second form, viz. 



1 1 w®(x + wcosu s/ — 1, y + w;sin«cos#'V / — -1, 

 Jo Jo 



z -f wsmu sinv 4/ — 1) sindudv 



j— I \ w x P(x + w cosu\S — 1, y + w smucosv\/ — l j 

 to> Jo Jo 



z + wsinwsinv V — 1) sinududv 



4ttW*=j 



...(k 



dw 



By a regular deduction of the integral from the equation 



•n / d . d 



\dw dw 

 or, putting 



dy dz' \ 



dw 



, . d . . d 1 



dx 



dy 



dz 



r,o, 



i£ +j£.+k—=T>, 



dx dy dz 



