Tuoxsox.— Glacial Action in Otago. 315 
By PARABOLA. By Survey. DIFFERENCES. 
- р hé aq 
Š 3 one (e) 4776 е 5207 43] 
Intermediate points | (f) 5130 uf 5251 qn 121 
At exit via aa (g) 3098 5598 кыы 000 
Thus, as will be observed by these differences, and the form of the 
parabola, the curve of the valley bed does not conform to the same. 
It was evident that the hyperbola would be yet more unconformable. 
The ellipse was then tried with the following results : 
—Let va the length of the valley, AB its rise, and 
ҮС the distance of any intermediate point from v be 
given, to find cp. Let A be the centre of the 
ellipse, then va will be the semi-axis major, AB the 
semi-axis minor, vC and cz will be abscisses to the 
ordinate CD... үл: BE::./vexez:cD. “Again, vz 
and BE being double of УА and AB, we have 525600:11196 :: ,/35400 x 490200 
12804, the ordinate required. 
= The other ordinates having been calculated in the same manner, afford the 
following comparison :— 
; By ELLIPSE. By Survey. DIFFERENCES. 
At source `... 54^ (a) :: 00 iis 000 5i 000 . 
( (b) 2804 - 96360* 1 © ay 821 
(c) 4212 4307 95 
At intermediate points < (d) 5012 4987 25 
(e) 5384 E. 5207 177 
(f) 5523 ie 5251 (vn 272 
At exit or e. (g) 5598 si 5598 : 000 
As the data of the actual survey are given for the lower terraces close to 
the river, I may state that these vary from 80 to 100 feet in difference of 
level from the river surface. The point marked with an asterisk, as stated 
before, is also given, not for the terrace, but for the ford, which accounts for 
so great a difference ; otherwise it will be seen that the actual curve of the 
bed of the Manuherikia agrees surprisingly with that of the ellipse, and, 
where it differs materially, such. deviations are in the positions to be expected, 
viz, where the eroding forces have had to expend themselves on the hard bars 
of schist rock already alluded to as crossing the valley. Here, then, we 
have in our first tried example a valley bed, whose length is 45 miles and 
difference of level 5,598 feet, conforming practically to the curve of the ellipse. 
That such might be a rule with other valleys was then surmised, though, no 
doubt, modifications would take place from peculiar contour of country and 
other natural obstructions. 
The next valley, then, which the data in the Survey Department enabled 
me to test was the Waitaki. The length of the Waitaki, from its source in 
the Mount Cook ranges to the sea, is 720,080 feet, and the altitude of the 
