THE CURVE ON ONE OF THE COORDINATE PLANES. 493 



§ xiv. 



To express the nine coefficients a 1} b x , c t , a. 2 , &c, which determine the posi- 

 tion of the system a, /3, y, we adopt for a the angle A which a forms with i and 

 the angle B which the projection of a on the plane (j, k) forms with j. Thus 



we have 



a = i cos A+jeosBsin A + Z; sin B sin A . 



For abbreviation's sake, we put 



cosA = a , sin A = a'; 



cjs B = & , sin B = &' . 

 Then comparing 



a = ia 1 +ja 2 + ka 3 , 

 a = ia +ja'b Q + ka'b' , 



we deduce 



a 1 = a , <x 2 = a'b , a 3 = a'b' . 



Calling /3 and y the directions forming with a a three-rectangular system in 

 such a position that /3 be coplanar with a, and i, we have then 



& = y a , 

 and y being perpendicular to i and to a we have 



C5y 6 i = , Sy a = , 



n 7o = Via 



y 2 =-l. 



- rfi = VHa = SHa - i 2 . a 2 , 



hence 



with 



Thus 



hence 



We put 



Thence 



This gives 



hence 



namely, 



— n z = a 2 — 1 = —a' 2 . 



n = a' (not = —a'). 



a'y = Via = ka'b —ja'b' 

 7o=-J b ' + kh o- 



Po = ( -ft + ^o)K + a'Oh + w y\ > 

 /3 = kb'a — i b' 2 a' +jb a — ia'b 2 , 

 i#o = — ia' +jb a + kb'a . 



We now turn the system a, fi , y round a as axis and to the amount of 

 an angle C, and we shall have 



&=p'P<fl , ,y=p'y 2', 

 where 



p' = cos|C + asin|C 



q'=cos^C — a sin £C . 



