702 
Proceedings of the Royal Society of Edinburgh. [Sess. 
which, he says, is seen to vanish on trying to find Laplace’s expansion of it 
in terms of minors formed from the last n -\- 1 columns and the minors that 
are complementary of those. Then, noting that the like outcome is not 
met with when the houndary-line necessary for the application of the said 
expansion-theorem is horizontal and bisects the determinant, he sets about 
obtaining the terms of the latter development in orderly fashion. Clearly, 
the first factors of those terms are all alike as regards their first n — m 
columns, but the remaining m columns may include another column of as 
or may not. Making a separation of the terms in accordance with this 
distinction, and calling the one aggregate S m _i and the other , where the 
suffix corresponds with the number of columns of b’s appearing in each 
first factor, and therefore also with the number of columns of a’s appearing 
in each second factor, Sperling gives evidence that is Sylvester’s 
expansion for | a u a 22 . . . a nn ! ■ | b n b 22 . . . b nn \ when in the formation of 
it there is an interchange of m-1 columns, that 2 TO is the corresponding 
expansion due to an interchange of m columns, and that the two 2’s occur 
with different signs. The conclusion is thus reached that, if we have 
previously proved the identity AB = 'E m _ 1 , the identity AB = must follow. 
It is important to note in passing that if Sperling had put zeros for 
a ln 3 a 2n , . . . , a nn in the second half of his 2^-line determinant, its value 
then would have been, when obtained in one way, ( — l) m_1 AB, and in 
another, ( — l) m_1 2 m . He would thus have made the natural extension of 
Cayley’s simple proof. 
LIST OF AUTHORS 
whose writings are herein dealt with. 
1839. Bianchi . 
1840. Chelini . 
1846. Terquem . 
1849. Hermite . 
1852. Salmon 
1853. Cauchy . 
1853. Saint Yenant 
1853. Hesse 
1853. Ohio . 
1853. Spottiswoode 
1854. Grassmann 
1854. Majo 
1854. Cayley 
1854. Brioschi . 
1855. Cantor 
1856. Heger 
PAGE 
. 676 
1856. SCHLOMILCH 
. 677 
1857. Rubini . . . . 
. 677 
1857. Bellavitis 
. 678 
1857. Baltzer . . . . 
. 678 
1857. Newman . . . . 
. 680 
1857. Del Grosso 
. 680 
1858. Janni . . . . 
. 680 
1858. Zehfuss .... 
. 681 
1858. Zehfuss .... 
. 683 
1858. Mainardi 
. 684 
1858. Gallenkamp . 
. 685 
1858. Sperling .... 
. 685 
1858. Casorati .... 
. 687 
1859. Salmon .... 
. 691 
1860. Sperling .... 
. 691 
PAGE 
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691 
692 
695 
690 
696 
696 
697 
697 
697 
698 
698 
698 
700 
701 
( Issued separately November 6, 1908.) 
