56 A. Mukhopadhyay — Hypokinetic Equations. [No. 1, 



Explanation of Plate I. 



Fig. 1. A piece of the placental cord of Zygcena blocMi, natural size. 



Fig. 2. Transverse section through the same, showing artery and vein, lym- 

 phatic (?) spaces, and three appendicala in oblique section with parts of two more 

 in vertical section. x 16. 



Fig. 3. A portion of one of the ajDpendicula of the same, showing the ramifying 

 vessel, x 21. 



Fig. 4. Transverse section through part of one of the appendicula of the same, 

 near its base, x 110. 



Fig. 5. Transverse section through uterine wall of Mi/liobatis nieuhofii, showing 

 fibrous and muscular coats, and mucous membrane, with the bases of three papillaa. 

 x 21. 



Fig. 6. Obliquely transverse section through part of one of the uterine papilloe 

 of the same, showing some of the simple follicles of the mucous membrane in oblique 

 section, and one of the racemose follicles. x 110. 



III. — On Ghbsch's Transformation of the Hydrolrinetic Equations. 

 By Asutosh Mukhopadhyay, M. A., F. R. A. S., F. R. S. E. 



[Received February 27th ;— Read March 6th, 1889.] 



A first integral of the hydrokinetic equations of Euler' may be 

 obtained by known methods in three cases: (1) Irrotational motion; 

 (2) Steady rotational motion ; (3) General rotational motion. It is the 

 object of this note to show how the method of applying Clebsch's 

 transformation to the third case can be materially simplified, and inci- 

 dentally the relation between the three solutions is pointed out.* 



Starting, then, with the hydrokinetic equations, we remark that 

 they may be at once reduced to the forms 



^ 2*4. 2^ + ^ = (1) 



at ax 



d " 2 wi+ 2uZ+f = Q (2) 



at ay 



| y -2^ + 2^+f = (3) 



■/ 



, ^+V+\q* 



g2 = u % _|_ v % _{. w % 

 * For the ordinary method, see Basset's Hydrodynamics, vol. i, p. 28. 



