182 Mr CHALLIS's RESEARCHES IN THE THEORY 



Then 



dr ' dx ' dy ' d% 



We proceed to shew next that the equation 



d^d) d^d) d^d> „ du dv dw 



zr^ + j^ + -j^ = 0, or J- + T- + ^- = 0, 

 daf df d%^ dx dy dx 



is satisfied by the kind of motion we have been describing. 



10. Let P (Fig. 1.) be the point whose motion we are considering; 

 Or, Nq, the focal lines to which the motion of the element at P is 

 directed. Let PNO be the straight line which passes through P and 

 the focal lines, cutting them in N and O. Suppose O to be the 

 origin of a system of axes, of which ONP is the axis of x, Oy coinciding 

 with the focal line Or the axis of y, and 0% perpendicular to the plane 

 yOx, the axis of %. The co-ordinates of P referred to another system 

 of rectangular axes AX, AY, AZ, are X, Y, Z: p is a point 

 indefinitely near to P, Pp is parallel to AZ, and the co-ordinates of 

 p are X, Y, Z+SZ: pqr is the straight line which passes through p 

 and the focal lines cutting them in q and r. Now let the equations 

 of Pp referred to the system Ox, Oy, 0%, be x = a% + a, y — b% + fi, 

 and the equations of pqr, x = dz-\-a, y = b'z + li'. Then 



„ l+aa' + bb' 



cos ^ Ppq = „ , — . 



Let ON=l, NP=r. Hence because Pp passes through P whose 

 co-ordinates referred to the axes Ox, Oy, Oz, are I + r, 0, 0, it follows 

 that l + r = a, and /3 = 0. Thus the equations of Pp become x = az + l + r, 

 y = bz. Again, because the line ^^gr passes through r, whose co-ordinates 

 are x — 0, z = 0, we have a' = ; and because it passes through q, whose 

 co-ordinates are y = 0, x = l, we have l=a'z, and = i's: + /3'. Hence 



a: = - = - -n, and consequently ft' = r. Thus the equations of 



pqr become x — a'%^ y^V% y . Also because Pp and pqr pass 



