MATHEMATICAL THEOBY OF STREAM-LINES. 
305 
Supplement to a paper on the Mathematical Theory of Stream-lines. By William John 
Macquorn Rankine, C.E., LL.B ., F.R.SS. Loncl. & Edin. 
Received January 8, — Head February 10, 1870. 
I. Addendum to § 1G . — Points of no Disturbance of Pressure. Mr. Berthon’s Log. 
— The points in the surface of the disturbing solid, and elsewhere, at which there is no 
disturbance of pressure, are given by the equation 
q 2 =u 2 +v 2 +w 2 = 1. (a) 
Such points can be found graphically for a given stream-line surface, by constructing a 
diagram such as fig. 5, and finding by trial the points for which AC = AB. 
At the surface of a sphere it is easily shown that we have 
sin 0 ; ............ (b) 
in which 0 is the angle made by a radius of the sphere with the direction of motion. 
Hence, on the surface of a sphere, the points of no disturbance of pressure are contained 
in the circle given by the equation 
0— sin -1 §=41' J 59' nearly. 
In February 1850 there was communicated to the Royal Society a paper by the 
Reverend E. L. Berthon, describing an instrument invented by him, called a “ hydro- 
static log and a more detailed account of that invention was read by Mr. Vaughan 
Pendred to the Society of Engineers on the 6th of December 1869. One part of that 
instrument consists of a vertical cylindrical tube, with a closed flat bottom, and having, 
in the front part of the cylindrical surface, near the bottom, a small hole, whose angular 
position, relatively to the direction in which the tube is moved through the water, is so 
adjusted that the pressure of the water outside produces no disturbance of the level of 
the column inside the tube. Mr. Berthon ascertained solely by experiment the “ zero- 
angle” or “neutral angle,” as it has been called, and found it to be 41° 30' — a result 
with which the theoretical value for a sphere agrees almost exactly. That agreement 
shows that the disturbance in the water caused by the short vertical flat-bottomed 
cylinder employed by Mr. Berthon was sensibly identical with that produced by a 
sphere, and also that, from the foremost points of the tube, as far round each way as 
the zero-angle, the disturbance of pressure was not sensibly affected by wave-motion, 
viscosity, or friction. 
II. Addendum to § 17 . — Interference of Waves. It was suggested to me by Mr. Wil- 
liam Froude, in a letter dated the 11th November, 1869, that one of the circumstances 
in the figure of a vessel on which the smallness of wave-resistance depends, is the inter- 
ference of waves originating at different parts of the vessel’s surface, so as wholly or 
