344 Prof. Sylvester's Investigation of the Wave-Surface, 

 Let the tangent plane to the wave surface be written 



cos «. . cos <^ , cos ^f ^ / \ifc 



. x-\- . vH a? = 1 (a.) 



v^ v^ ^ ' v^ ^ • 



d 



cos <w + , COS (p _ COS %!/ 

 ' d'——- a-— — 



then t'i . „ . '" 



(/ . COS CO rf . COS 45 rf COS tj; ^ . . 



-Ij^-'^^TW '^^W^ = ' ^'-^ 



then = " (y) becomes A ?J7 + B>j3/ + C?;s = (1.) 



and=»(/3) -y--J:+ + -— ;s = ... (2.) 



and = " a) may be written under two forms, viz. 



(a^-O ASx + {b^-v,^) B>j3/H-(c^-V)C?^ = 1 (3.) 



From(l.) A?^ + B>jj/= - C?^ (5.) 



From (2.) A y ^ + ^^ = - ^-^ ^ (6.) 



From (2.) and (1.) A{a^-c^)£ x + B {b^-c^)Yii/ = J (7. 

 From (3.)and(4.) A{a^-c^-)^ + B(Z>^-c^)^ = c^ (8.) 

 From (3.) and (6.) C^ c^ s^ - B* ^^^ - A^a^a;^ 



= ABxJ,(a^| + i^l) (9.) 



• In lieu of v, we might write v^^ in the denominator without affecting 

 the result. 



,,hatE£i_^=yit:.!)^-!i^ 



t Ob.er,e..hat_- = V.l;i^_/_ _,,„,,,„„„ fo, th, „3,. 



