Illustrations of Symmetrical Integration. 349 



is 



l l J_ 



a™- 1 n\^ w ~ 1 y m ~ l z m ~ l J ov ' J 



2. The method employed in solving the primary type, in ex- 

 amples of this species, suggests the possibility of the instanta- 

 neous integration of such partial differential equations as 



p cos 2 x + q cos 2 y = k cos 2 3 ; 



or, more generally, taking three independent variables, 



„ dw t » dw t c> dw % t \ 



cos^ a? . t- + cos^ 5 v . -j- + cos^ 2 . -j- — k cos z w; : . . (a) 

 <to «y «* 



or, again, such equations as 



{a*-x*f . ~ + (b*-ff . ~ + (c*-z*f . ~ = k(m?-w*f; (0) 

 or 



(l + ^).-J + (l + ^).J+(l + ^).J=*(l+^). . (7) 



These equations, in fact, are respectively reducible to the forms 

 d . tan w d . tan w d . tan w 



d . tan x d . tan y d . tan z 



= *, . . . («') 



a. sin 2 — a. sin x — a. sin -1 — 



« + S + !?«*, . m 



a. sin" 1 - «.sin _1 7 a.sm -1 - 

 a b c 



d.tan _1 w d.t&n~ 1 w cLtan _1 w_, . ,. 



dA2xs- x x rf'.tan- 1 ^ "^.tan- 1 * ~~ 5 ' V'J 



the solutions of which equations are respectively 



k 

 tan w = ^ (tan a? + tan y + tan z) + w (e tan *, e*" 1 *, e tan *), .... (a") 

 o 



^11-^=1(^11-^ + sin- 1 ! + siii-^)^^^"^, ^ sin - 1 ^ ^ - 1 "), m 



tm~ 1 w=^{tm- 1 x+tm- l y+t3in- 1 z)+u {e t&n ~ lx } e t&n ~^ f e tan_1 *). ( 7 ") 



3. Let it be proposed to integrate the partial differential 

 equation 



"T" — a • 



x y y 2 

 The solution of this equation is, by the ordinary operational 



