218 Wisconsin Academy of Sciences^ Arts and Letters. -^ 



b,„ V and b,, z parallel to Pc. Then right angled triangle b^„ vC has angle 

 C b„, V equal to 5— v + (n:±z); and right angled triangle b„ z has angle 

 C b„ z equal to :&+u— (a:±z). Produce line b„ k to u, b,,zu is a right angled 

 triangle. In triangle u C k angle C k u equals 90' + a, angle k C u equals 

 z and C u k equals 90°— (a:±z). 



^. , , A, sin 5 sin z 

 Side u k : 



cos (a ± z) 



„ ^ A, sin S sin z 



b„ u = B, cos 3^ + —^ -. — --^— 



" ' cos(ar±z) 



b„ z = B, cos 3- cos (6i:±z) + A, sin 3 sin z. 



Likewise, 



b,„ v=B, COS 5 cos {a ± z) — A, sin 3 sin z. 



In triangle b„ z C, 



B, cos 3- cos ((.r±z)+A, sin 3 sin z 

 cos [3+u— (cr±z)]= - ■ = — 



In triangle b,,, v C, 



B, cos 3 cos (u:±z) — A, sin 3 sin z 



cos [3 — v4-(ar±z)]= 

 Let E-': 

 A=B, 



cos [34-u— (a+z)] 



__a;^— e; 



B, cos 3 [ cos ( (t-±z)+]/14-£'- tan 3 sin z 



B„ 



Let l^=cos (n- it z), and n=^l+£^-tan 3 sin z. 



For mass {vi cos 3- cos^ a) at a,, in direction Pc on particle P. 



m cos- a A-„ B, cos'- b cos^ 3 (1+n) m A/ B, cos^b (1 + n) 



Attraction 



A„^=: 



D- A,^ B„ cos'^ 3 cos'^ a. D'A;-B„ 



A 2 B,2 cos^o: 



B„* cos^b 



m B,^ (1 + n) cos'^ a 



Att., etc., then= ^^^rrr^^-^ • 



U- B/ 



Likewise for mass {m cos 3 cos'^ or) at a, „ in direction Pc on particle P 



A +f —'"^ ^'° (1 " '^) cos- or 

 ^«. ^3^ 



Draw C X perpendicular to b,, u, and draw a line from C perpendicular 

 to b,„ j produced. In right angled triangle b„ X C, or in a right angled 

 triangle similar having hypotheuuse B„ C instead ©f b„ C, 



Side for C X = A, sin 3 cos a. 



Side for b„ X = B, cos 3- A, sin 3 sin x 



