44 



given, and its power as a working instrument is illustrated by ex- 

 amples. By the help of this symbol, certain expressions which occur 

 in endeavouring to extend the theory of symmetric products to 

 quintics are greatly simplified and presented in an intelligible form, 

 and the direct calculation is effected of a certain sextic equation on 

 the solution whereof that of the general quiritic may be made to 

 depend. 



The third and concluding section is chiefly occupied with the cal- 

 culation of the perfect symmetric product for the quintic. By com- 

 bining Eulerian with Lagrangian functions, and introducing a simple 

 artifice, the symmetric product for the quintic wanting in its second 

 term is obtained. This result is then made the basis of a calculation 

 for the perfect form. Employing the property of seminvariancy 

 pointed out by Mr. Cayley, the author succeeds in effecting the cal- 

 culation of the symmetric product for the complete quintic. This 

 product is composed of three hundred and twenty-five functions of 

 twenty-four dimensions. 



III. " On the Surface-condensation of Steam." By J. P. 

 JOULE, LL.D., F.R.S. Received Oct. 10, 1860. 



(Abstract.) 



In the author's experiments steam was passed into a tube, to the 

 outside of which a stream of water" was applied, by passing it along 

 the concentric space between the steam-tube and a wider tube in 

 which the steam-tube was placed. The steam-tube was connected 

 at its lower end with a receiver to hold the condensed water. A 

 mercury gauge indicated the pressure within the apparatus. The 

 principal object of the author was to ascertain the conductivity of 

 the tube under varied circumstances, by applying the formula sug- 

 gested by Professor Thomson, 



where a is the area of the tube in square feet, w the quantity of 

 water in pounds transmitted per hour, V and v the differences of 

 temperature between the inside of the steam-tube, and the refrige. 

 rating water at its entrance and at its exit. The following are some 

 of the author's most important conclusions. 



