80 Mr DE morgan ON THE GENERAL EQUATION OF 



in which the factors equal to unity, and introduced for symmetry, 

 have the brackets []. Develope these expressions, from which we 

 obtain the following equations: 



fl„=p^+y^+jo"% l,= qr-^qr' + q"r", 



h^=q' + q'' + q"\ m,= rp^-t'p' + r"p" (17), 



Co = r^ + /" + r"*, n„=pq +p'q' + jo'Y'. 



These, added together, the three last having been respectively multi- 

 plied by 2 cos I, 2 cos rj, 2 cos ^, give from (16) 



«o + *o+Co + 24 cos f + 2»?o cos »? + 2«„cos ^=3T\ 

 The first side of which, developed from (5) and (6) gives 3 V^* whence 



T=y/Vo (18). 



If the process by which (17) w^as obtained from (2) be repeated 

 upon (17), that is, if at,ha-lo, Wana—a^la, &c. be formed, we shall have 

 equations of a similar form, substituting instead of p, p' &;c. such functions 

 of them, as they themselves are of a, y3, &c., the first sides of the equations 

 being from (7), ^o ^^ *^^ ^^^^ three, and V^ cos f, F^ cos ri, Vg cos ^, in 

 the last three. These equations are such as would arise from sub- 

 stituting in (2), 



^ ^ ,^ — instead of a y~ — and y-^ ^ for a and a", &c...(19), 



which are therefore the values of a, a', &c. in terms of p, q, &c. 



From (1), by means of (14) and (18), can be deduced the following : 



^/YgX=px'+p'y'-!t-p"^, 



VT,y = qaf + q'y'+q"fi (20), 



-v/Fo as = r x' + / 2^' + r"%', 



and the equations of the axis of x', referred to the oblique axes 

 X, y, and k, are any two of the three, 



qx-py — 0, ry — q% = 0, p%-rx=Q (21), 



