342 Prof. Sylvester's Notes to his Analytical Development 



cos 60 = — ^ 



"" ^/a'^x'^h^f^-c'^z^ 



Let 4/a*^2 + ^*/ + c*;s^ = P, 



then P^^ = u^ (sec co)^ 



Now let a, /3, y be the angles of inclination between the 

 coordinate planes and the front in which the line of vibration 

 lies, and A. some quantity to be determined. I have shown in 

 proposition (3.) 



that if A. cos a, ■=: {a} — v^) x 



then will A cos /3 = (h^ — v^)y 



and X cos y — (c^ — t;*) z 



- ^2v''{a'x^ ^ b'^f +c^z^) 

 + t;4 



= P^ ~ v\ 

 Again, 



/Jcosjt)^ (cosj)^ (cosy)n _ ^s . ^. . ^. _ j 



_ j __. (cos g)^ (cos /3)^ (cos y)^ 



Now m 4 = TT ~\o. 4^ = -X (cos Oif, 



p2_u4 u* (sec co) 2— I?* tr ^ ^ 



And in Mr. Smith's investigation of the form of the wave sur- 

 face (already alluded to*) by great good fortune 1 find ready 

 to my hand 



(cos a)2 (cos /3)^ (cos yY _ 1 



(r) being the radius vector to the point whose tangent plane 

 is parallel to the point in question. 



Hence (cot c)^ = ^-^,-— ,j = ^;,-^, 



_ P' 



(p) being the length of the perpendicular from the centre 

 upon the tangent plane for p = v. 



Hence (cot oo)^ = the square of the cotangent of the angle 

 between radius vector and normal. 



• Sec p. 78, 261, and 335. 



