196 PROFESSOR A. E. H. LOVE ON THE 



Again, when v = %, the equation can be written 



3+-W3- V^l^-z^r^**^} = 0, 

 or \ ar or/ I Jo J 



or 



where the left-hand member is certainly positive when a^ < 1 ; also the differential 

 coefficient of the left-hand member with respect to x is 



and this expression cannot vanish for any value of x which is greater than unity. 

 Hence the equation (80) cannot have more than one positive root. 

 24. Now take v = 0, and write the equation (80) 



When y? = 4, we have 



-r 2 -r 4 



2 



3 3.5 V 33.5 



44123 4 7 



3. 5. ..13 3.5...15V 177 3. 5. ..19 

 and 



1 _ 1 _ 45045 

 a?-l" 3 3. 5. ..13' 



-I)-..., 



Hence, when y? = 4, the sign of the left-hand member of the equation (81) is plus. 

 When X* = 5, we have 



. ..1-1+ > 



3 3.5 3 3.5 



20056 { . 15625 L 5 



Also 



3.7.9.11.13.3 3.7.9.11.13.3.17V 19 



5 8 / 5 \ 



f 3. 7. 9. 11. 13. 3. 17. 19.21 \ ~23/ + "" 



15625 Ii> 677, 



17 19 

 and therefore 



5 . 5 2 20733 



3 3.5 3. 7. ..13. 3' 



but 



i(3.7...l3.3) = 20270 + < 20733. 



Hence, when x 2 = 5, the sign of the left-hand member of equation (81) is minus. 

 It follows that the value of x?, or tfo?, which satisfies the equation is between 4 and 5. 



