382 Sir W. R. Hamilton on a Mode of deducing 



X 2 Y 2 Z* /. i 



+ -*r + -7? = x > • • • • (*•) 



b* ^ c 2 



XrfX YrfY , ZrfZ : /0 , 



-^- + -^ + -^ 2 - ==0 » ' * (2 ' } 



^X + i/Y + 2Z = ) (3.) 



.r<ZX+j/riY + zrfZ = o, . . . (4.) 



XtfX + YrfY + ZdZ m 0, . . (5.) 



« S +/ + « 8 =X 8 + Y 2 + Z 2 ; .... (6.) 



between which we are to eliminate X, Y, Z, and the ratios of 



their differentials. 



3. The equations (1.) and (2.) are satisfied by assuming 



X = a sin cos <$>, Y = b sin sin <$>, Z = c cos ; (7.) 

 and then the equation (3.) gives 



tan = ~~i =— ; : 5 • • • • (8.) 



a x cos <p + b y sin <J> 



while the comparison of the two values of tan —tq» deduced 



from (4.) and (5.), gives 



ax cos $ + by s'm<p — cztan _ a 2 cos $ 2 -|-& 2 sin$ 2 — -c 2 



a x sin § —by cos $ (« 2 — & 2 ) sin $ cos $ " } * * *' 



and the equation (6.) becomes 



(x* + j/ 2 + * 2 ) ( I + tan 8*) = (a 2 cos <f> 2 + b* sin <J> 2 ) tanfl 2 + c 9 . (1 0.) 

 It remains therefore to eliminate and <p between the three 

 equations (8.) (9.) (10.). 



4. Substituting for tan 0, in (9.) and (10.), its value given 

 by (8.), we easily obtain 



A tan <p + B cotan $ m C; (I.) 



A'tan<f>-fB' cotan $ = 0; . . . f . (II.) 

 if we put for abridgement 



A = {c^-b^abxy; 

 B = (a 2 — c 2 ) a b x y ; 

 C = (6 2 -c 2 )« 2 ar 2 +(c 2 -« 2 )Z» 2 3/ 2 +(i 2 -a 2 )c 2 ^ 2 ; 



r 2 = x 2 + y 2 + s 2 ; 

 A' = r 2 (Z> 2 y + c 2 ^ 2 ) - c 2 A 2 (j/ 2 + 2 2 ) ; 

 B' = r 2 (a 2 .r 2 -f c 2 * 2 ) - c 2 a 2 (.r 2 + s 2 ) ; 

 C = - <2 (1* - C*) a b x y. 

 And eliminating <J> between the equations (I.) and (II.), we 

 find: 



(AB'~A'B) 2 + (AC'-A'C) (BC'-B'C) = 0; (III.) 

 a form for the equation of the wave, which we have now only 

 to develope and depress. 



