Appleby South Junction (the southern link between the North Eastern and Midland lines) required additional legislation and this was provided via section 4, subsection 10 of the Midland Railway (Additional Powers) Act, 1875, which received Royal Assent on 29 June 1875.
The relevant text is as follows:
4. Subject to the provisions of this Act, the Company may make and maintain, in the lines and according to the levels shown on the deposited plans and sections thereof, the railways and alterations of railways herein-after described, with all proper stations, sidings, approaches, works and conveniences connected therewith, and may enter upon, take, and use such of the lands delineated on those plans and described in the deposited books of reference relating thereto as may be required for that purpose.
The railways herein referred to and authorised by this Act are:-
...
10. The Appleby South Junction (3 furlongs 2 chains 40 links in length), situate wholly in the parish of Saint Michael, Appleby, otherwise Bongate, in the county of Westmoreland, commencing by a junction with the Eden Valley Railway of the North-eastern Railway Company, and terminating by a junction with the Railway No. 2 authorised by the Midland Railway (Settle to Carlisle) Act, 1866, now in course of construction: