61 
of Edinburgh , Session 1866-67. 
which is easily seen to agree with (3). From this many well-known 
theorems may be deduced at once. 
(5.) By (1) and (2) we have 
S. (Va/3 + V/3 7 ) (Yfiy-^Yya) (\ T ya-^Yal3) 
= 2 S . Ya/3 Y/3y Yya = — 2 (S . afiyf . 
(6.) Similarly, if we put 
$ = V(Va(3+V/3y Y/3y+Yy a), e = V(Vf3y+Vya Yya+Yafi), 
£ = V (Yya + Va/3 Ya/3 + \ T /3*y), 
we have 
S . <5e£ = — (S • Va/3 +V/S 7 V/2 7 +V 7 a V^+Va/3 ) 2 
= — 4 (8 . a/3 7 ) 4 . 
Such theorems may be multiplied indefinitely ; but we have 
already reached a stage in which the algebraic form is very 
unwieldy. 
(7.) The common rule for the multiplication of two determinants 
is, of course, easily found by this process in the form 
S . a/3y S . a (3 y [ 
Saa A S a/3 L 8ay i 
$(3a S S/3% 
S7 a i s 7& s 77. 
The following Gentlemen were admitted Fellows of the 
Society : — 
T. B. Johnston, Esq. 
George F. Barbour, Esq., of Bonskeid. 
David Davidson, Esq. 
Peter Waddell, Esq. 
George Harvey, Esq. 
George Stirling Home Drummond, Esq. of Ardoch. 
Professor Fuller, Aberdeen. 
’ The following Donations to the Library were announced: — 
Transactions of the Boyal Scottish Society of Arts. Vol. VII. 
Part 2. Edinburgh, 1866. 8vo. — From the Society. 
