4-22 Mr. Tovey's Researches in the 



this case the sum 5! . \I/ (/•) A x^ A f/ A ;: (art. 3) will vanish: 

 for let the plane of x and y pass through the molecule m and 

 divide its sphere of influence into two hemispheres; then, 

 since the arrangement of the molecules will, by the supposi- 

 tion, be sensibly the same in both, it follows that the terms of 

 this sum will be half of them positive and half negative, and 

 will destroy each other. 



(5.) If we denote the length of the waves -^, , — ,— , by 



A, X,,\^; and their velocities —, -^ , -jr^ "y '^> ^'/» ""ii' 

 the equations (3.) of the paper at p. .500, give 



(6.) Since our object is only to ascertain the forms of the 

 wave-surfaces, we will, for the present, neglect the terms in 

 these equations which depend upon the lengths of the waves, 

 and suppose Vj = s^, v^^ = s^^ ; then, by the formulae at p. 271, 

 we have 



»,2 = |- 2 . ((^ (r) + ^Kr) A/) A a:% 



m (^-^ 



v,^ = — 2 . (<f (r) -h 4' {r) A z') A x' . 



Now, let the axis of z coincide with that of z, , and let d 

 be the angle formed by the axes of x and x^; then, when 

 X, r/, z, and x^,y,i,z^, have the same origin, and are coor- 

 dinates of the same molecule, we have, by the principles of 

 analytical geometry, 



X = x^ cos 9 — j/i s\n &, 



J/ = Jtr, sin 6 + i/, cos d , 



- = z,; 

 and consequently, 



A a; = Ax, cos 9 — At/, sin 9 , 



Ai/ = A :r^ sin 8 + A;y^ cos fl , 



Az = Az,, 

 The last equations give 



A x* = A x^ cos^ $ + Ai/,^ sin- fl - 2 A x, A j/^ sin Q cos S , 



Af=Az;;_ 

 and if we substitute these values in the second of the equa- 

 tions (3.), we have 



