7Q Art. 4. -M. Kuniyeda 



From this we can easily deduce 



I cos n sill - -^r- 



./ \ // / // 



4 cos ^V- '- ^ ' ^ ^ ' ^ \ I y \ > / } 



These tormulae hold respectively for 



-1 < V < 1, 

 and for — 1 <; v <; 2, 



it being understood that, in certain special cases, the expression 

 of the rigiit-hand side must be replaced b^^ its limits. For v = ^ 

 thej^ assume the forms 



' / "^cos u COS f^-^ — = iv^a-) (-sin 2,9 + cos 2,? + e-^^), 



./ \ U J 's/u - - 



/"*cos n sin { ^'\ ^-'' = hVßr:) (sm 2,9 + cos 2ß-e--^) ; 

 ./ \ u J V ' 



and their values are expressible in terms of elementar}" functions 

 also when 



^ = - i- 4 



(the last value, of course, only in the second integral). 



Now write ^w for ß\ and put u = kr in the integrals. We 

 obtain 



/"cos Kx COS (-'"-) 4^^ = 5 , ( ™ f' r_/'(2v'(W.)} 



+ J-''[2V(m;0] -e-^''''V''[2/V(m;0] +ei'''='V-''[2V(w/'0]1 



/^os^^sinf^'L)-^!, = ,--V-(Tf r^^i^VlW.)] 

 ./o \ a- / a:. 4 cos |-vn- \ -^. / L 



Now, when /J is large, 



