1889.] A. Mnkhoi^adhysij— Elliptic Functions and Mean Values. 229 

 and 



0,2 -c;2 (i2_,.aj2 ^1 



Making these substitutions in (12), we get 

 ^ _ Trabc 



6 



ac i2_y2 f X 



_ ^ ^ i2p (_^'2,2,e)-,.3Pi (-/32c'2,«) i 



(i2_c2Na.„2_„2)2 i ^ '^5' 



(i2_c2)a(„2_ 



and since , 



Ydr, 



we have, by substituting for V and eliminating M between this equation 

 and (25), a remarkable relation connecting four definite integrals. 



§ 6. Second Case. 

 We now proceed to the consideration of the second case where only 

 two of the vertices of the ellipsoid, viz., the extremities of the longest 

 axis, are outside the intersecting sphere, so that we have 

 a y r y b y c. 



It is not necessary to repeat the whole of the previous calculation for 

 this case, as by Prof. Catalan's beautiful transformation,* it may be 

 made to depend on the preceding investigation. Thus, if we put 

 X = ax', y - by', z = cz' 



- = c, -j = b', - = a', 

 aba 



equations (1) and (2) are transformed into 

 We have also 



c' Z 1, b' y I, a' y I 

 c' Ah' A a' 

 a' y h' y \ y c'. 



* Problemes do Caloal Integral par B. Catalan, Joiiinul de Mulhematiquils 

 (Liouvillc), scr. i, t. vi (1841) pp. 4,10—44.0 ; cX. p. 439. 



