200 REV. R. HARLEY ON SYMMETRIC PRODUCTS, AND 
= (22000) + 2(21001) + (20002) + 2(12100) + 2(10012) 
+ (02200) + 2(01210) + (00220) + 2(00121) + (00022). 
Consequently 
7°= 5 § (44000) + 4(43001) + 6(4:2002) + 4(41003) 
+ 8(33101) + 4(32102) + 4(382012) + 4(22211)} 
+ { (22000) + 2(21001) + (20002) + 2(12100) 
+2(10012) + (02200) + 2(00121) + (00022) }. 
Developing, reducing,* &c., we are conducted to 
7°= >’ [ (66000) + 6§ (65001) + (61005)? + 15§ (64002) 
+ (62004) t + 20(63003) + 24(55101) + $36(54102) 
+6(54012) + 6(52104) + 36(52014) } + 244 (53103) 
+ (580138) } + 4 (44220) + 28(44202) + (44022) t 
+ $20(44211) + 2(44121) + 20(44112)} + 84 (43203) 
+ (43028) } + 40(431138) + $4(43221) + 24(43212) 
+12 (43122) + 12(42218) + 24(42123) + 4(41223) ¢ 
+ 6(42222) + 16§ (33321) + (83312) } + 452(33222) 
+ 8(382822)%]. 
Applying the theorem in Art. 29 we have 
¥7°=383'(66) + 63'(651) + 153'(642) + 203'(633) 
+ 24:3/(5511) + 213'(5421) + 245'(5331) 
+ 803) (4422) + 423'(44.211) + 83'(4382) 
+ 403 (43311) + 403 (48221) + 863'(42222) . 
+ 963'(33321) + 303 (383222). 
By the aid of the formule in Art. 34, and the values of 
[1], [2], &c. in Art. 33, we find 
> (66)=8-5'Q4, 3(651)=— 3-5'Q*, B(642) =-3-5*Q' 
(633) =3-5'Q*, (5511) =0, 3(5421)=0, 5(5331)=0, 
> (4422) =0, 3(44211)=0, 3(4332)=0, =(43311)=0, 
* It may be convenient here to illustrate the principle of reduction pro- 
eeeded upon in this Paper in dealing with circular functions. According 
to the notation of the text, represent 3’eta® 2¥ > a= by 3/(aByde); and 
suppose (e.g.) that, of the five exponents, + is the greatest ; then =’ (aByée) 
must be replaced by its equivalent 3'(7dea8). Or, suppose that the greatest 
exponent (7) is repeated, and the function takes the form 3’(ayyde); this 
must be replaced by its equivalent =/(yydea). Andsoon. This mode of 
reduction is uniformly followed in the text. The greatest exponent leads. 
