1889-90.] Prof. Rutherford on Fibre of Crab and Lobster. 149 
segment causes them to return to their normal length after contrac- 
tion, but does not lead to the appearance of the proper clear seg- 
ment . , which is only seen when the fibrils are forcibly extended to 
their full physiological length. The phenomena of contraction are 
essentially due to vital changes occurring in the tissue between the 
equator of Bowman’s element and that of the intermediate segment. 
Some Multinomial Theorems in Quaternions. By the 
Rev. M. M. U. Wilkinson. Communicated by Professor 
Tait. 
(Read April 7, 1890.) 
The formulae established in a preceding paper are particular 
instances of some very general formulae to which I wish to draw the 
attention of mathematicians. 
A. On Notation and Symbols employed. 
Suppose (r + s) vectors taken, , a 2 , . . . a r+s , and from these 
any selection of r vectors made. Calling the product of these 
vectors taken in order (that is, so that two vectors a m , a m+n , never 
occur in the order a m+n a m , but always in the order a m a m+n ), p, and 
the product of the remaining s vectors, likewise taken in order, q f 
we form the four quaternions, 
SpS#, S.pYq , YpSq, Y.pYq , 
if the sum of the suffixes in p is congruent to Jr(r+ 1), modulus 2 
(that is to say, if the sum is odd or even according as the sum in 
what will be always regarded as the first term in the series is odd 
or even), the sign of the quaternion will be + , if otherwise, - . 
Four series are thus formed, each series containing C(r, s) terms, 
C (r, s) standing for the number of combinations of (r + s) things 
taken r together. We define, as follows, 
S(r, s) = S ± SpSq ; 
Z(r,s) = 2± S.pYq; 
Y(r,s) = 2± Y.pYq; 
B(r, s) = % ± Y.pSq . 
