THE CURVE ON ONE OF THE COORDINATE PLANES. 495 



we see that to pass from an expression in which # g , 6 g , c. 2 enter alone (exclusive 

 of 0s, # 3 , e a ) to the similar expression in a s , b s , o 3 we have only to change 



b into b', and b' into ( — b ) . 

 First 



u 2 = Sa^^ «= a 2 « 2 + a' 2 (6 2 c 2 + c 2 c' 2 ) 



which is independent of b , V. Then 



i; 2 = 2a 2 <x 2 2 



1> 2 = a?a' 2 b 2 4- b 2 (a Q b e — b'c') 2 



+ c\~a \c'-b'c ) 2 , 



v 2 = b 2 [a 2 a' 2 + b 2 a 2 c 2 + c 2 a 2 c' 2 ] 

 + 2& ?>'[ — b 2 a c c' + c 2 a c Q c'] 

 + b' 2 [b 2 c' 2 + c 2 c 2 ]. 



cos2B = B 

 sin2B = B\ 



namely, 



Introducing 

 we have 



2V = 1 + B 

 26' 2 = 1-B . 

 By analogy we introduce also 



cos2C = C ,sin2C = C', 



but we will leave c 2 , c' 2 as they are, putting only 2c c r = C. Thus by multiply- 

 ing both members of the expression of v 2 by 2, 



2v 2 = ( 1 + B )[> 2 a' 2 + a 2 { b 2 c 2 + c 2 c' 2 )] 

 + (l-B Q )[b 2 c' 2 + c 2 c 2 ] 

 -B'(& 2 -c> C. 



2t> 2 = [« 2 a' 2 + a 2 (b 2 c 2 + c 2 c' 2 ) + (b 2 c' 2 + c\ 2 )] 

 + B [aV 2 + a 2 (b 2 c 2 + c 2 c' 2 ) - b 2 c' 2 - c 2 c 2 ] 

 -B'(6 2 -c 2 )« C. 



By the above-mentioned change of b into b', b' into — b we get 



B = b 2 -b' 2 changed into b' 2 -b 2 = -B , 



B' = 2b Q b' changed into — 2b'b = — B' . 

 Putting 



[a 2 a' 2 + a 2 (b 2 c 2 + c 2 c' 2 ) + b 2 c' 2 + c 2 c 2 ] = x 



B |>%'2 + « 2 (6 2 c 2 + c 2 c' 2 ) - (6V 2 + c 2 c 2 )] - B'(& 2 - c> C = x , 

 we have at once 



2v 2 = (x +x) 



2w 2 = (x — x). 



