Dr T. Muir on the Theory of Determinants. 
517 
SCHERK (1825). 
[Mathematisclie Abhandlungen. Von Dr. Heinrich Ferdinand 
Scherk, . . . iv. + 116 pp. Berlin. Pp. 31-66. Zweite 
Abhandlung : Allgemeine Anflosung der Gleichungen des ersten 
Grades mit jeder beliebigen Anzahl von nnbekannten Grossen, 
iind einige dahin gehorige analytische Untersuchnngen.] 
The only previous writings of importance known to Scherk were, 
according to his own statement, those of Cramer, Bezout (1764), 
Vandermonde, Bezout (1779), Hindenburg, and Eothe. His style 
bears most resemblance to Rothe’s, whose paper, however, he does 
not speak of with unmixed eulogy, characterising it as containing 
^‘eine strenge aber ziemlich weitlauftige Anflosung der Aufgabe.” 
The main part of the memoir consists of a lengthy demonstration, 
extending, indeed, to 17 pages quarto, of Cramer’s rule, or rather of 
Cramer’s set of three rules (iv., v., iii. 2), by the method of so-called 
mathematical induction. The peculiarity of the demonstration is 
that it is entered upon without any previous examination of the 
properties of Cramer’s functions (determinants) j and it is note- 
worthy on two grounds — (1) as being new, and (2) because the 
properties, which it really if not explicitly employs, had also not 
been previously referred to. 
The cases of one equation with one unknown, two equations 
with two unknowns, three equations with three unknowns, are dealt 
with in succession, the solution of one case being used in obtaining 
the solution of the next. All three solutions are noted as being in 
accordance with Cramer’s rules, and the said rules being formulated, 
and supposed to hold, — for n equations with n unknowns, it is 
sought to establish their validity for n-\-\ equations with ?^ + 1 
unknowns. In other words, the set of n equations being 
nn n-ln-\ n-2n—2 11 
ax a X + ax ax = s 
111 11 
nn n-ln-l n-2n-2 11 
ax + ax a x +.... + ax = s 
2 2 2 2 2 }- 
71 n 71-171-1 n-2n-2 11 
ax + ax + ax +.... + ax = s \ 
71 n n 71 71 j 
