AS EXPLAINED BY THE HYPOTHESIS OF FINITE INTERVALS. I6l 



that velocity. And for the same reason , must be either in the direc- 

 tion of transmission, or making a constant angle with it, and as the 

 introduction of a constant cannot in any mannfr affect our^llts, w^ 

 may consider , and ^ respectively as defining the place and velocity 

 of the wave at the end of the time t. 



Another remark is also important, that since from the constitution 

 of the medium it is indifferent in what direction the axes of co o" 

 dmates are taken, all the functions which we may introduce iLw 

 at on" "^ T T '"^^P-^-^-t of e, and ^, so that we might 



at once suppose the direction of transmission to be the axis of y ai d 

 mh for ,,, this, however, I shall not do, as it does no !ppe ' 

 necessary, and it is convenient to retain the symbol ,, on other accoL 

 to be noticed hereafter. The above remark wiU be mainly xiX " n 

 pointing out to us what are the quantities to be rejected in our 

 equations of motion. 



Let us now take as the solution of equation (1) the form we have 

 obtained from equation (2), which is perfectly aUowable, since the latter 

 IS only a particular case of the former: the quantities n and k are of 

 course not necessarily the same for both. 



Put the solution under the form 



« = « cos (ci—kp) ; 

 .-. ca = a cos {ct-kp-Jfh(j)-u cos (Ct-kp) 



= -a cos (ct-kp) . (1 -cos Hp) + asm (ct-kp) sin kSp, 



where Sp = S^- cos e + ^y cos ,^ + ^. cos >^ is the projection of the distance 

 PQ on the line OP. 



By the substitution of this expression for Sa, and analogous ones 

 tor 6fi and ^7; our equation (1) becomes 



dF = ~ 2«2^(^) sm^ -y- + « sin {ct-kp)^cp{r) sin k^p 



+ 2 — — bx { - 2a sin^ -J±+ asm (ct-kp) sin kSp] 

 Vol. VI. Part I. X 



