20 Mr. Mac Cuutacu on the dynamical Theory of 
SECT. Il.—LEMMAS. 
Lemma I. Let a right line making with three rectangular axes the angles 
a, B, y, be perpendicular to two other right lines which make with the same axes 
the angles a’, 6’, 7 and a”, B”, 7’ respectively, and which are inclined to each 
other at an angle denoted by @; then it is easy to prove that 
sin @ cos a = cos f' cos 7’ — cos B’ cos 7’, 
sin 6 cos B = cos 7 cos a’ — cosy’ cos a’, (A) 
sin 6 cos y = cosa’ cos B’ — cosa’ cos f ; 
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supposing the first right line to be prolonged in the proper direction from the 
origin, in order that the opposite members of any one of these equations may 
have the same sign, as well as the same magnitude. 
If the last two right lines be perpendicular to each other, we have sin 6 = 1, 
and the formule become 
cos a = cos f’ cosy’ — cos B’ cos 7; 
cos B = cos cos a’ — cos 7’ cos a’, (B) 
cos y = cos a’ cos B’ — cos a” cos B’ ; 
but in this case the three right lines are perpendicular to each other, and there- 
fore we have, in like manner, 
cos a’ = cos fp” cos y — cos Bcos 7’, 
cos B’ = cos 7’ cos a — cos y cos a’, (3’) 
cos y' = cos a’’ cos B — cos a cos B” ; 
and also, 
cos a’ = cos B cosy’ — cos f' cos 7; 
cos 6’ = cos y cos a’ — cosy cos a, (B’) 
cos 7’ = cos acos ’ — cosa’ cos Bp. 
The last three groups of formule will still be true, if we suppose the first 
right line to make with the axes the angles a, a’, a’, the second the angles 
B, B’, 6’, and the third the angles y, 7; 7’. 
