AS EXPLAINED BY THE HYPOTHESIS OF FINITE INTERVALS. 159 

 fir ' 2 '^ dx" 2 





sii"^ + £i"'+U.V^+ 



and similar expressions for S(3 and ^7. 



But it is evident that tlie sum of a series of terms of which one 

 factor m each is <p{r) and the other ^^^ ^y". ^.., where m + n^pis an 

 odd mteger, will be identically equal to zero ; since if ,n, for instance is 

 odd, we shall have, for any particular vahies of ;•, S,j, S., two equal 

 values of ^^, the one positive, and the other negative, and one of the 

 quantities m, u, p must be an odd number. Substituting, therefore the 



*: :::■:,: °/e„t„T; :: ;. ^''"'™ <"• -^ --""-^ ■"-««- 



and again 2 . (/■) ^^'- = 2,^(^-)^y^=2<^(r)^s^ = 2«^ 

 writing w= for abbreviation; 



• — = «= /^ 4. ^^'« , d'a \ 

 ■ ■ elf \dx' ^df^ -del 



dt' - W ^^^""lle] 



dt- ydx-" ^ dtf dz' j ' 



three equations of remarkable simplicity and elegance; of which the 

 following are evidently solutions: 



a = a cos k {ni-(ex+/i/+g-z)\, 



/3 = ft cos/c {nt-{ex +fy+gz)\, 



y = c.cosk {nt-{ex+fy+gz)}, 



subject to the restriction that e-+f^+g.'. = i. We can easily get rid of 

 this restriction by writing cos for e, cos <p for/ and cos ^ for o- 



