104) BOOK OF MATHEMATICAL PROBLEMS. 



tion in x, y of the second degree. Thus the straight lines joining 

 the origin to the points 



ax 9 + by 9 + c + 2ay + 2b'x + 2c'xy = 0, 



ay -f b'x + c = 0, 

 are c (ax 3 + by' + 2cxy) = (a'y + b'xf. 



The results of linear transformations may generally be ob- 

 tained from the consideration that, the origin being unaltered, 



x 3 + 2xy cos <u -f y* 

 must be transformed into 



X s + 2XT cos ft + 7 a , 



if (x, y), (JT, T) represent the same point, and w, 12 be the angles 

 between the axes. Thus if 



ax* + by* + c + 2a f y + 2b'x + 2c'xy 



be transformed into AX 3 + ......; and if we give h such a 



value that 



h (x* + y* + 2xy cos<o) - {ax* + ....... } 



separate into factors, the corresponding transformed expression 

 must also separate into factors. Hence the two equations 



- c (h- a)(h-b) + 2a'b'(hcos( 1 >-c')-(h -a) a* 



and -c(h-A)(h-)+ ......................... . .................... =0, 



must coincide : and thus all the invariants may be deduced. 



One form of the equation of the circle may be mentioned : if 

 (x lt y } ), (x a , y a ) be the extremities of a diameter, the equation of 

 the circle is 



( x _ Xi ) ( x - ag + (y - y,) (y - y a ) = 0. 



530. The equation of the two straight lines which pass 

 through the origin and make an angle a with the straight 

 line x + y = Q is 



2xy sec 2a + y s = 0. 



