258 Proceedings of Royal Society of Edinburgh. [sess.. 
just as if he had expanded the determinant according to products of 
the elements of the principal diagonal. 
Interrupting the process of solution for a moment Jacobi draws 
attention to the fact that elegant relations between the sixteen 
quantities a , a , a" , a" , . . . and the sixteen A , A' , A", A"' , . . . 
have been handed down by Laplace, Vandermonde, Gauss, and 
Binet, — an interesting remark as showing what writings on deter- 
minants were then known to him. Upon the subject of these 
relations, however, he does not enter, contenting himself with 
giving two sets of equations derivable from them with the help of the 
sixteen results 
a — - JcA , /5 = -&B, .... 
The first set resembles the half-score of equations obtained near the 
outset, being 
2 2 ft 2 , ,2 
— a +a/ + a + a ' =/r, 
— a (3 + a/5' + a"/3" 4 - a" /5'" = 0 , 
— y8 + y'8' 4- y"8” + y'"8'" — 0 . 
The other set consists of sixteen of the type 
a/5' - a'/5 = - (y"8"' - y"'8")e , 
where e = ± 1 , and is in effect a prolix way of stating the fact, 
nowadays familiar, that any two-line minor of | a/3'y"8'" \ differs 
from its complementary minor only in sign, if it differ at all. 
Further he inserts at this stage the reverse substitution of that 
with which he started, viz., 
- 8' + a cos P + /5' sin P cos 0 + y sin P sin 0 
COS if/ ~ g _ a cog P _ gi n p C0S Q _ y gi n p gi n 0 * 
. f f - 8" + a" cos P + /5" sin P cos 6 + y" sin P sin 0 
smi/rcos <f> - g _ a cosP - p sin P cos ^ - y sin P sin 
- 8"' + a"'cos P + /5"'sin P cos 0 + y'"sin P sin 0 
smi/fSin</> — g _ a cos p _ p s i n p cos 0 - y sin P sin 0 ’ 
to which is added the fact that the common denominator here is 
the quotient of k by the common denominator in the original 
substitution. These results, he states, are easily proved, — doubtless 
