of Edinhurgk, Session 1882-83. 
lol 
ill embryo vertebrates, is regarded as being not its original and 
proper duct, but a secondary connection which has been formed with 
a lost sense organ, placed at or in front of the anterior end of the 
pharynx, and homologous with the dorsal tubercle in the Tunicata. 
In conclusion, Ussow and Julin have conclusively shown that 
the dorsal tubercle is not merely a sense organ. The complex 
structure which the tubercle usually presents seems to indicate that 
it is not merely the aperture of a duct. Whether, as I suggest, it 
may be a sense organ into which the duct has come to open can 
scarcely be determined on the evidence at present in our hands. 
The lines of investigation which may be reasonably expected to 
throw additional light upon the matter are — (1) The exact course 
of development of the neural gland and the dorsal tubercle, and 
further information as to the pituitary body ; and (2) The examina- 
tion of the condition of the gland and its ducts throughout the 
Tunicata, and especially in a large number of specimens of Ascidia 
mammillata^ a species in which these organs appear to be in a 
variable and highly interesting condition. 
5. On the Quaternion Expression of the Finite Displacements 
of a System of Points of which the Mutual Distances re- 
main Invariable. By Gustave Plarr, Docteur ks-Sciences. 
Communicated by Professor Tait. 
When we try to define, from a purely geometrical point of view, 
the rigidity of a system of points, we are inclined to consider the 
invariability of the mutual distances of the points a sufficient defini- 
tion. But when we take this definition as a starting point, in the 
problem of expressing the finite displacements of the system, we 
soon discover that the invariability of the distances is not sufficient 
for insuring the invariability of the relative positions of the points. 
In this paper we have, with the help of the quaternion-method, 
transformed the fundamental condition of the problem into an 
equation which presents two separate factors. One of the factors 
solves the problem by representing the displacements as a screiv- 
rotation of the system. The other factor gives the geometrical per- 
version of the figure which the system possessed before the displace- 
ment of its points. 
From the expressions of the two kinds of displacements, we are 
