284 
MR. MACQUORN RANKINE ON AXES OF 
Note referred to at page 261. 
On Sylvestrian Umhrce. 
Without attempting to enter into the abstract theory of the Umbral Method, it 
may here be useful to explain the particular case of its application which is employed 
in this paper. 
Let U be a quantity having an absolute value, constant or variable (such, for ex- 
ample, as any physical magnitude), and u,v, . .. &c. a set of quantities, m in num- 
ber, such that U is of them a homogeneous rational function of the wth degree. 
There are an indefinite number of possible sets of m quantities satisfying this con- 
dition ; and the quantities of each set are related to those of each other set by m 
equations of the first degree, called equations of linear transformation. Let 
Wo Wl, 
W2J Wj, 
be two such sets. 
Let denote the coefficient of mV, 
in the development of 
and let 
(M-hM-f....)”, 
= WaV*....}. 
The two sets of coefficients A,, are connected by linear equations of trans- 
formation, the investigation of which is much facilitated by the following process. 
Let two sets, each of m symbols, Kj, /3i, &c. ... jSg, ... &c. be assumed, such that 
= ci^u^ + / 32^;2 + 
and that, consequently, 
(a,M, -l-/3,Vi +....)”= 2 { 4 , .. .. 
= (a2M2-l-^24^2+ • • • •)“= 2 { 4 . . . M 2 V. 
Then if the m equations of transformation between the two sets of symbols Mj, /3,... 
and 0 ^ 2 j fie formed, and if from them be deduced the equations between the two 
sets of products and &c., and if, in the latter system of equations, 
there be substituted for each product a“/3*..... the corresponding coefficient A „,4 , 
the result will be the system of equations sought. Also, if any function of the pro- 
ducts be invariant {i. e. a function, whose value, like that of the original 
function U, is not altered by the transformation), the corresponding function of the 
coefficients A will be invariant. 
The symbols «, /3, &c., with reference to their relation to the coefficients A, are 
called umhrce', that is, factors of symbols, whose equations of transformation are simi- 
