Royal Society. 549 
far as is practicable, not only above the general mass of iron, but also 
above any smaller portions of iron that may be in its vicinity; or such 
portions of iron should be removed altogether. 
4th. The steering-compass should never be placed on a level with 
the ends either of horizontal or of perpendicular bars of iron. 
5th, Theextreme ends of an iron vessel are unfavourable positions, 
in consequence of magnetic influences exerted in those situations. 
The centre of the vessel is also very objectionable, owing to the con- 
necting rods, shafts, and other parts of the machinery belonging to 
the steam-engine and wheels, which are in continual motion; inde- 
pendently of the influence exerted by the great iron tunnel in this 
part of the ship. 
6th. No favourable results were obtained by placing the compass 
either below the deck, or on a stage over the stern. 
7th. It was found that at a position 203 feet above the quarter- 
deck, and at another 134 feet above the same level, and about one 
seventh the length of the vessel from the stern, the deflections of the 
horizontal needle were less than those which have been observed in 
some of His Majesty’s ships. 
The author proceeds to point out various methods of determining, 
by means of a more extended inquiry, whether the position above 
indicated, or one nearer to the deck, is that at which the steering- 
compass would be most advantageously placed. 
The concluding section contains an account of some observations 
made by the author on the effects of local attraction on board dif- 
ferent steam-boats, from which it appears that the influence of this 
cause of deviation is more considerable than has been generally ima- 
gined; and he points out several precautions which should be observed 
in placing compasses on board such vessels. 
“ Researches on the Integral Calculus. Part I.” By Henry Fox 
Talbot, Esq., F.R.S. 
The author premises a brief historical sketch of the progress of 
discovery in this branch of analytical science. He observes that the 
first inventors of the integral calculus obtained the exact integration 
of a certain number of formule only ; resolving them into a finite 
number of terms, involving algebraic, circular, or logarithmic quan- 
tities, and developing the integrals of others into infinite series. The 
first great improvement in this department of analysis was made by 
Fagnani, about the year 1714, by the discovery of a method of rec- 
tifying the differences of two arcs of a given biquadratic parabola, 
whose equation is += y. He published, subsequently, a variety of 
important theorems respecting the division into equal parts of the 
arcs of the lemniscate, and respecting the ellipse and hyperbola; in 
both of which he showed how two arcs may be determined, of which 
the difference is a known straight line. Further discoveries in the 
algebraic integration of differential equations of the fourth degree 
were made by Euler; and the inquiry was greatly extended by Le- 
gendre, who examined and classified the properties of elliptic inte- 
grals, and presented the results of his researches in a luminous and 
well-arranged theory. In the year 1828, Mr. Abel, of Christiana, in 
