Aft. -i i.} TIG 9XOMMTBT xil. 



tlien the addition <.f equations (3) and (4) 



gives A' + B' = A + B (since sin 2 + cos 2 = 1). 



Again, A' - B' - 2J7sin20 + (;l -)cos20 . . 

 hence 



(4' - #) + 4 77' 2 = j (-4 - B? + 4 //*! (sin 2 20 + cos 2 'J f > ), 



(X-^ + 4//-. . . 



t .. , ^4' 2 - 2 ^ ff + #' 2 + 4 7/' 2 = ^ 2 - 2 AB + ^ + 4 7^. 

 But by (6), 



^l f2 + 2 ^'J?' + ^ /2 = A* + 2 4# + & ; 

 hence, by subtraction, 



JT* - ^ B = JJ* - ^IJ5, ... (0) 

 and the function 7/ 2 AB is also unchanged l>y tin 1 t 

 formation of coordinates, through the angle 6. M 

 if a transformation of coordinates to a new origin be j >!- 

 formed as in Art. 179, A, B, and h are not cliai 

 nor, therefore, the functions A + B and H 2 AB. Surh 

 functions of the coefficients, which do not vary when the 

 transformations of Arts. 175 and 179 are performed, are 

 called invariants of the equation for those transformations. 



If, as in Art. 175, 6 be chosen so that 



. . , (10) 



.A. ~~ AJ 



then II' = 0, and equation (9) becomes 



- A'B' = J5P - AB. . . . (11) 



' 77 



Again, from eq. (10), sin 20 = - 



V(^i By 



and cos 20= A ~ B 



hence, equation (8), A'-B'= 2 H . . . . <T2) 



sin 2 



Since sin 20 is positive (Art. 17", >. therefore, the. 

 A' B' is the same as the sign of //. 



