324 REPORT— 1873. 



and fifteen more precisely similar formulte for 



F, Q, &e. (p. 287). . . . : 



Putting the arguments u and ((' = 0, we have : — 



'Bj'' = f — u^ —v' — w', and similarly for is-', is" ^ 'm'"; ~> 



I 

 Tc"'=2tu—2vw, and similarly for p"2, „'' ; 



'k"''^=2tu-\-2vw, and similarly for p'"^, o-'"^ ; 

 Tc, Tc", p, p', ff', cr" vanish. 



■ From these we easily deduce the following : — 



llli '4 1114, , 4 /;4 . lilt 



la ■CT =p -f- (7 =p + (T , 



'"I "4 "M I 7 '4 4 I 7 '"4 



V 



2 _ ^'^2 '2 "2 2 '"2 27 '"2 "27 '2 '"2 2 ' 



'Zij-'Cr — ^ 'UT ^=. (y (J y 'UJ li ^ /c =zp O" , 



2 '"2 '2 "2 7 '"2 2 2 '"2 "''^ 2 7 '2 "2 



(1) 



> 



(2) 



• • (3) 



with many similar formute (p. 288). 



From formulte (1) we easily see that we have an expression of the form 



P==aP'= + /3S'HyP"= + gS"^ 



by putting the arguments u, u' equal to zero and the quarterperiods, we 

 determine a, fi, y, S, and we find 



with similar formulae for S^ P'"^, &c., also in terms of 



P'^ S'\ V"-, S"- (4) 



Gopel next investigates the relations which exist between the products 

 PS, P'S'. By means of Table 1 he proves very easily that such relation must 

 be of the form 



aPS+ 6P'S' + cP"S" -I- dV"'S"' + eQR+/Q'R' + *7Q"R" + hQ"'R"' = ; 



and then, by the help of the second Table, he proves that this equation gives 

 rise to the two following: — 



aPS +cZF"S'" + eQR +7»Q"'R"' = 0, 



ftP'S' + cF'S" +/Q'R' + (7Q"R" =0. 



By putting the arguments u, u' = 0, and also, making use of equations (3) 

 of this section, we obtain the following two equations derived from the second 

 of those we have just written down : — 



k"'p"'Q'U' = ^"V"'FS'-«rffP"S", 



F"p"'Q"'R" = - ^trP'S' + ^"V"T"S". 



Squaring the first of these equations and making use of equations (4), 



