Formulce for sn 8u, c?i8u, dn 8u, in terms ofsnn. 481 



find sn Su= ^Q^^ 



S 4 -fc 2 P*' 



8a 



_ S 4 — 2P 2 S 2 -ffc 2 P 4 

 S*-fc 2 P* 



, q S 4 -2k 2 P 2 S 2 + £ 2 P* 

 dn 8%= ! 



S 4 -£ 2 P 4 



§ 2. The numerators of sn 8u, cn Su, dn 82^, and their common de- 

 nominator, may therefore be deduced by combining linearly the four 

 expressions PQRS, P*, S 4 , P 2 , S 2 , where 



{4 - (8 + m)x* + + (8Jfc*+ 8& 6 > 10 -4& 6 a5 12 } , 



Q=l-8^ 2 +(8 + 20^ 2 )^-(24/^ + 32/v 4 )^ + (54^+16Z; 6 > 8 



- (24k 4 + 32k 6 > 10 + (8^ + 20k 6 > 12 - 8&V 4 + 

 R= 1 - 8 W + ( 2 0k 2 + 87; 4 > 4 - (327^ + 24/j 4 )^ + (1 6k 2 + 547j 4 >8 



- (32&*+24& 6 > 10 + (20^ 6 + 8^ 8 )^ 12 -8^ li + /^ ? 



8=1 - + (32& 2 + 32Z; 4 ) a; 6 - (1 6Z; 2 + 58** + 1 6#> 8 



+ (32^ + 32Z; 6 )^ -20/ 4 ;^ 12 + / 4 ;V 6 , 



and aj denotes, as throughout this paper, sn u. 



The values of PQRS, P 4 , S 4 , P 2 S 2 were calculated in the following 

 manner : 



The squares P 2 , S 2 , and the products PS and QR, were first formed. 

 Then P 2 , S 2 were multiplied together, and the square of PS was 

 formed : the agreement of these two results verified the values of P 2 , 

 S 2 , P 2 S 2 . The expressions for P 4 and S 4 were then obtained by 

 squaring P 2 and S 2 ; these calculations being performed in duplicate. 

 To obtain PQRS the expressions for PS and QR were multiplied to- 

 gether; and as a verification the product PQ was formed, and this 

 was multiplied successively by S and R. 



§ 3. The resulting formulae are shown in the following tables, in 

 which the mode of arrangement is almost obvious : thus, for example, 

 the numerator of sn 8u. 



{8 



-(80 + 80& 2 > 2 



+ (192 + 968k 2 + 192k 4 > 4 



- (128 + 2496& 2 + 2496k 4 + 128Z.- 8 )* 6 



+ (1 728k 2 - 741 6k± + 1 728k 6 ) a; 8 



+ (87984k 4 +879347 i ; 6 )a3 10 



+ (80k 2 8 + 80k 30 ).r 58 

 -8WJ 



