1058 HON. LORD M'LAREN ON SYSTEMS OF 



more contour lines of the homogeneous surface in x, y, z, which is represented by the 

 equation. Examples of these are given in the sequel. 



Other applications of the combination of soluble functions of x and y, or r and 0, 

 will readily suggest themselves. The following may suffice as illustrations : — 



(1) Let the equation consist of powers of quantities (x 2 + y 2 ) and (x 2 — y 2 ) as 



(x 2 +y 2 ) n + A 1 (x 2 +y 2 ) n - 1 + A 2 (x 2 + y 2 ) n - 2 + + • - . = A n (x 2 -y 2 y . 



This is equivalent to 



r 2 +A 1 r 2re - 2 +A 2 r 2 "- 4 ++. . . = r**co8»'2fi . A„ , 



whereby is determined in terms of r, and thence x and y. 



(2) (x l +y m ) n + A 1 (x l +y m ) n - 1 +A 2 (x l +y m ) n - 2 + + = A n (x'-y m )p . 



Take ii l = x l ; v 2 = y m ; and solve the resulting equation in u and v, which is of the above 

 form. Then u and v are found from r and 0, whence x and y are determined. 



(3) This solution is evidently capable of extension to any function in the form on 

 the left side of the sign of equality, where the quantity on the right side can be 

 expressed as the power of a cosine or sine of a multiple of 0, or a soluble function of 

 such a sine or cosine. In these equations each term is a homogeneous function, and the 

 solution depends partly on this circumstance. 



(4) If the right-hand term consists of a power of x alone or of y alone, the equation 

 is solved by writing r p coa p for x p , or r p sin p for y p . 



(5) If the left side of the equation consists of powers of the quantity (x l — y m ), and 

 the right side of a single term x p or y q , or (x l + y m ) p solutions of r can be found to the 

 argument 0, and thence u x v 2 and xy are similarly found. 



8. To find the Condition under which parallel Sections of a given Surface may be 



similar Curves. 



The condition is evidently fulfilled if one of the quantities x, y, z be given explicitly 

 in the equation. The surface may be conceived as traced by the motion of a generating 

 curve controlled by a guiding curve. 



Suppose the generating curve to be a homogeneous function of x, y, equated 

 to a function of z only. Then, as the sections parallel to XY are to be similar 

 curves, the generating curve must move parallel to itself with varying parameter, 

 and so as always to touch a guiding curve in the place XZ. Let the equation of the 

 guiding curve be of the form 



x n - z n ± B^i 1 ± B 2 z n ± 2 d= . • • + B^iP = . 



Then the equations of the generating and guiding curves are 



f(xy) = x" + A 1 x n - 1 y+A 2 x n - 2 y 2 ^=FA„y n =p .... (1). 

 f( x y) = x s " - z y " ± B^"* 1 ± . . . + B h z^p = . . . . (2). 



