318 



whei e 



REPORT 1873. 



Now let 

 then 



M =.^,, „ (v, w) f,, „ (v', tv') 0„, „ (v", w") <^„, „ (v'", w'") 

 —f2. 1 (^> ^<^) ^2. 1 (»''' *«'') 00. 1 (:>■'"> ^"") 00, I (^"'' ^"')» 



M' =.^3, {y, W) 03, C*^'' ^^') 01, (^"' ''''") 01, (^"'' «^"') 

 -03, 1 (^' '*'') 03, 1 (^'> '''') 0b 1 (^"' **'") 01, 1 (^"'' '*'"')' 

 Ml" =00, (^l' ^^l) 00. «^ O 02, (^l". W'l") 02. «, <') 



-00, 1 K' '<^i) 00, 1 (^^ '^i') 02, 1 K"' <') 02, 1 «". *^"'). 



Ml"'=01. K' *<'l) 01, (^'.'' '"/) 03, «', '^") 03, («'l"'. <") 



-01, 1 (''P ^i) 01. 1 (^'' «'i') 03. 1 O'l"' ^.") 03. 1 K'"' •^'i'")- 



/ /\ f r\ II III II III 



v=iV =-y), tc=w =U, V = — v , w = — %u , 



02.1=0, .?)3.,=0, <^„,,=0, 0i,„=O, 

 aud the equation becomes, suppressing the accents, 



<S>\. 0'o, (^ «') + 0'3, 0\, (^' "') = 0'o, 0\, (^'» «') + 0'l, 1 0^3, 1 (^, '^)*, 



whence 



;^_ 0^0 0'l.oK"') I 0\ 0'2. (^> ^^) I 0'l. 10'.3. iC'^,^^') . ' 



0\, 0\ (l'> »'') 0\. 0'o, i^^ '0 0\ 0\ (^. ^<^) ' 



and similarly from 9>d and 12(i, 



1___ 0'3, 3 0'l,o(^^) , 0\3 0'2.oK«') I 0'l. 1 0^3, 2 C'^, ^) ^ ^^^ 



0'2, 3 0'o, iy, ^) 0\, 3 0'o, (^'' «') 0'2, 3 0^0, (" > ^)' 



0^3, 2 0'., (^, ^) ^_ 0%. 2 0'2, (^. ^') , 0'l, 10\3O'>'»^) 



0'2, 2 0\oK«') 0%, 2 0'o, K '«') 0\2 0\„O',«')' ^ 



In like manner we obtain from the Table, by causing the argument to vanish, 



0*3, 3 - 0*0, = 0*2. + 0*3, 2 = 0*0, 2 + 0*2, 3' 



0*3, 3-0*1, 1 = 0*3, + 0*3, 2 = 0*0, 3 + 0*2, 3' 



0*3, 3-0*2, 2 = 0*3, + 0*0, 2 = 0*0, 3 + 0*2, 0' 



0^3, 0^3, 3 = 0\, 0\, 3 + 0°o, 0'o, 3, 



0^3, 0^3, 2 = 0% 0%, 2 + 0'o, O0'o, 2' r 



0\, 2 0^3, 3 = 0^2, 2 0'2, 3 + 0^, 2 fo, 3' ^ 



with twelve similar equations, which will be found on p. 417. 



* Because ^3, ^{v, w)——(p^^ i(~*'> —'"')' wliich may be proved thus. It is seen at once 



CO QO '^ „ 



that S,.^(n) = 2„0(-M). Hence ^3, ^{v, w) = j:,„p'"'e--"''-e^{w-2mA, q), 



(B) 



(C) 



-GO 



-OD 



-00 



00 



03, i(-«. -ui) = -s:,„p">6-"'%{-w-277i.A, 9)= -03, ,(t', w). 



-co 



